Differential volume computer



H. S. FLlCK April 5, 1960 DIFFERENTIAL VOLUME COMPUTER 3 Sheets-Sheet 1 Filed Sept. 19, 1955 April 5, 1960 H. s. FLICK E 2, 3

DIFFERENTIAL VOLUME COMPUTER Filed Sept. 19, 1955 v 3 Sheets-Sheet 2 NEW/47d? J lrwiefi 5. 1 2/616 IN VEN TOR.

4 ZOE/V64 Abril 5, 1960 H. FLICK 2,931,565

DIFFERENTIAL VOLUME COMPUTER Filed Sept. 19, 1955 3 Sheets-Sheet 3 Had/I80 5'. /Z/C INVENTOR.

United States PatentO DIFFERENTIAL VOLUME COMPUTER Howard S. Flick, Lakewood, Califi, assignor to BJ Service, Inc., Long Beach, Calif., a corporation of Delaware Application September 19, 1955, Serial No. 534,993

7 Claims. Cl. 235-61) This invention relates to an improvement in mechanical computers. It is particularly useful in connection with computers using ball and disc type integrators.

It often happens that a solution is desired to an equa tion involving the difference between the squares of two quantities, i.e., (x a By factoring the expression within the parenthesis it will be seen that the solution of the equation is proportional to the expression (x+a)(xa). The present invention takes advantage of the equivalency of the latter expression with the former to solve equations involving (x -a In addition, by varying the setting of circuit constants or integrator combinations, other quadratic or higher order equations can also be solved by the computer.

Although the invention has general utilization in the solution of equations involving the difference between the squares of two quantities, it has special application in computations of annular volume between two generally concentric surfaces. Specifically, in the drilling of oil wells it is necessary to know what the annularvolume will be between the walls of a bore hole and the exterior surface of a string of casing to be cemented into the hole. It is necessary to compute this volume accurately, since it will determine the quantity of cement to be used. If

I the computation is incorrect in that it is too low, too little cement will be pumped into the annular space between the casing and the bore hole walls. Not all of the drilling fiuid will be displaced by the cement, and the upper portions of the casing will not be properly cemented in place. Conversely, if the computation is too high, an excessive amount of cement will be used, and the job will be more expensive than necessary due to wasted cement. p

Prior to beginning a cementing job in the past, it had been customary to lower a caliper into the bore hole, and to measure the bore hole diameter at selective intervals. An average diameter was then calculated. Since the outer diameter of the casing, and the depth to which the casing was to be set into place were known, the differential volume could be calculated. However, this method was obviously inexact since the arithmetical average bore hole diameter should not be used; rather the root mean square of all the instantaneous bore hole diameters should be used.

p The inaccuracies of past computations are avoided by the instant invention, since it provides a continuous integration of the instantaneous bore-hole diameter. The computer may,- for example, comprise two ball and-disc type integrators linked together in series. The computer may be usedto'determinethe total bore hole volume rather than differential volume'if desired. This would be done by feeding bore hole diameter data (D into the first integrator from a bore hole caliper. Inaddition, the first integrator would receive data (M, representing height above the bottom of the hole. The output of the first integrator would be such that it would be proportional to -/zx'D By passing this output to the second integrator, and there superimposing identical D information, the output of the second integrator would be proportional to hXDB By a suitable adjustment of circuit constants this output could be made to actuate a meter reading bore hole volume directly in cubic feet.

For the performance of a cementing job, the quantity desired is the differential volume between the casing and the bore hole, rather than the total bore hole volume; means are therefore provided to subtract out the casing volume.

The formula for the differential volume desired is:

where V =differential volume D =varying bore hole diameter D =constant external diameter of the casing H :height of the bore hole above the bottom of the hole dh=unit of height I The last equation above cannot be solved mathematically, because h cannot be expressed in terms of the arbitrarily varying D However by actually measuring the instantaneous D adjusting it according to the constant D and correlating it with h, as a caliper is pulled up through a bore hole, the true differential volume will be obtained. As will be explained below, the sum D +D is fed into one integrator and the difference D D is fed into the second integrator. By feeding the output of the second integrator to a meter, suitably adjusted, the differential volume can be read directly. If desired, a continuous log can be kept, correlating h with V It will also be seen from the description that follows, that by simple adjustment the computer can be made to solve other quadratic equations such as result from the product of (xa)(x+2a), (x-2a)(x+a), (x+a)(x+a), (x-a)(xa), and other similar expressions. Still further by the use of more than two integrators cubic, quartic, and higher order equations can also be solved.

Accordingly it is an object of the invention to provide a computer which will solve equations of the type which are the product of 'a plurality of factors of the type (aria), where x is a variable and a is a constant.

It is also an object of the present invention to provide a computer which will solve equations involving the difference between the squares of two quantities.

' It. is another object to provide acomputer which. will continuously integrate the difference between the squares of avariable and a constant.

It is a further object to provide a computer which can determine the differential volume between a cylinder of constant volume and a generally cylindrical surface enveloping it, the surface having the same axialldimension as the cylinder but with a varying diameter cross section. I

It is a further object to provide .a logging method and apparatus whereby the differential.volume between the :walls' of a bore hole and can be'm etered.

Referring now to the drawingsz Figure 1 is a pictorial showing of a caliper which is being drawn up through a borehole to provide a continuous indication of changing bore hole diameter.

Figure 2 is a schematic showing of the computer assembly. h v Fi r f Ri .YisW 9!...ths. prnputen casing to. be set within the hole Figure 4 is a sectional view of the diameter informa tion adjusting carriage, taken along the line 4-4 of Figure 3.

Figure 5 is a sectional view of the carriage taken along the line 5-5 of Figure 4.

Figure 6 is another sectional view taken along the line 6-6 of Figure 4.

Figure 7 is a view of the casing diameter setting knob, taken along the line 7-7 of Figure 3.

Figure 8 is an enlarged showing of the switch contacts illustrated in Figure 3, but with the contacts open. Referring now to Figure 1,. it will be seen that the caliper 13 is drawn up from the bottom of bore hole 14 by means of line 15. The caliper is of a conventional design having extending legs 18 which flex according to the changing bore diameter, D As the legs flex, a varying electrical signal is generated. This signal is passed up a conductor carried within the line 5, and used to actuate the computer mechanism illustrated in Figures 2 through 8. it should also be noted that Figure 1 illustrates in phantom the bottom of a string of casing 20 which is to be cemented in place within the bore hole. The outer diameter of the casing 20 is indicated by the symbol D It will be seen that in order to determine the quantity of cement necessary to set the casing 20 in place, it is necessary to accurately determine the annular volume between the casing 20 and the bore hole 14. This volume will depend on the varying differential diameter between the bore walls and the casing, as well as the longitudinal quantity h as shown in Figure 1. This 11 information may be obtained by means of a direct takeoff from the shaft 16 to which sheave 17 is keyed. Or the shaft 16 can be connected to a first Selsyn motor which generates a signal to rotate a second Selsyn motor in synchronisrn with it. This is the preferred arrangement and, as seen in Figure 3, the second Selsyn motor is designated by the numeral 40. By means of the gears 42, 43, Selsyn motor 49 causes disc 44 of Integrator 1 to rotate at a speed proportional to the rate of increase in it (see Figure 2).

It should be noted that the zero datum level shown in Figure 1 is not the bottom of the bore hole 14. Rather it is the depth to which the bottom of the casing will extend. Knowing this depth in advance, the line 15 can be payed out to this depth, taken up to account for stretch and the length of the caliper, and the zero datum level thereby set with reasonable accuracy.

The varying bore hole diameter data D measured by the caliper 13 actuates a first Selsyn motor (not shown), and synchronous impulses are transmitted to a second Selsyn motor 19, carried within the computer (see Fig. 3). The Selsyn output shaft 22 will therefore rotate in one direction as D increases, and in the opposite direction when D decreases. By means of the bevel gears 24 and 25 this motion is effective to rotate shaft 27,

which is journalled within support piece 28.

As can best be seen in Figures 3, 4, and 6, shaft 27 is externally threaded, so as to mate with internal threads within the generally squared shaft 29. Shaft 29 changes in cross section as viewed from right to left in Figure 4. From its end face 30 to shoulder 31 it is squared, but with the corners rounded off (see Fig. 6). From shoulder 31 to end face 26, shaft 29 is cylindrical. Snap ring rides in a peripheral groove adjacent end face 26, and is effective to retain shaft 259 within shaft 32. Shaft 29 is restrained from rotation, when shaft 27 turns within it, because the squared portion of shaft 29 is mounted within the square bushing 36. Rotation of shaft 27 is therefore effective only to move shaft 29 axially. It should also be noted that since the corners of the squared portion of shaft 29 are rounded (see Fig. 6), the overriding shaft 32 can be freely rotated.

Shaft 32 is also of changing cross section, being cylindrical in the areaof the threaded portions 45 and 46.

As seen in Figure 5, this changes to a square cross section, and remains so from shoulder 47 to end face 48. The squared portion of shaft 32 fits within a square opening cut out of gear 52. Shaft 32 can therefore move axially of, and be rotated by gear 52. Gear 52 has a central extension sleeve 54 which is mounted within support member 56, and is held in place by snap ring 59.

It will be noted that two transversely extending arms 33 and 34 are carried on shaft 32. Arm 33 operates ball race positioning arm 37 of Integrator 1, and arm 34 operates ball race positioning arm 33 of Integrator 2. In the schematic showing of Figure 2 arms 37 and 38 are shown as being of one piece construction, whereas actually they consist of parts 37a, 37b and 38a, 38b as shown in Figure 3. The entire assembly of shafts 27, 29 and 32, and arms 33, 34, 37 and 38, forms a carriage 39, which moves ball races 71 and 73 simultaneously, and in an amount proportional to the changing D data being fed in from Selsyn motor 19 and shaft 22.

During a bore hole logging operation the arms 33 and 34 move in unison, but their relative distance apart does not change. However, prior to commencing the log of the bore hole, it is desired to set their relative positions so that arm 33 will reflect D +D and arm 34 will. reflect D D This is done very simply. Inasmuch as threaded portions 45 and 46 are oppositely threaded, portion 45 being left handed, and portion 46 right handed, rotation of shaft 32 will move each of the arms 33 and 34 toward or away from each other by an equal amount. The distance these arms are from the center point 49 of shaft 32 is proportional to the predetermined casing diameter, D The arms are set by rotating shaft 32, which is turned by the worm and gear drive 51, 52. The worm gear 51 is operated by shaft 50 and control knob 55. A suitable step-down gear arrangement is made a part of the scale against which control knob 55 rotates so that inner dial 57 is rotated one complete revolution before outer dial 58 moves over one unit (see Figure 7). The scale is such that the reading is in a 4:1 ratio with respect to the casing diameter factor, D Thus if the control knob were set at 20 as shown on the dial in Figure 7, the computer would subtract out a casing volume corresponding to a D of 5".

Referring now to Figures 2, 3, and 8, it will be noted that control shaft 60 can be moved inwardly against the action of spring 61 so that the bevel gear 63 engages gear 25. The shaft 60 is manipulated by means of control knob 64 in order that the ball race positioning arms 37 and 38 can be'manually set at their zero starting position. The zero starting position for arm 37 is one wherein the ball race 71 is at the center of the inte grator disc 44; the zero starting position for arm 38 is one wherein the ball race 73 is at the center of the integrator disc 53. These zero positions are both attained simultaneously, and at the time that control knob 55 is set to read zero. This is done with exactness since arms 37b and 38b can be adjusted with respect to arms 33 and 34 by means of the screw threads 80, 81 and the set nuts 83, 84. The adjustment of arms 37b and 35b is communicated to the arms 37a and 38a respectively by the abutting connection therebetween, and the spring and nut ties 86, 87, 89, 90.

It will be observed that when bevel gear 25 is rotated manually, the shaft 27 will. be rotated and will in turn axially move shaft 29. When the left end of the shaft 29 is moved sufiiciently to the left it will be adjacent the switch contact 66. As it moves farther it will separate switch 66 from contact 67 as shown in Figure 8. When this occurs, the light 69, which had been turned on when the computer was initially energized, will be extinguished. The operator of the computer would then know that the ball race positioning carriage had been set at its starting position.

Referring to Figure 2, it is seen that the computer includes integrator l and Integrator 2 linked together 49 of shaft 32 are equal and proportional to D in series connection. The integrators'are preferably of the ball-and-disc type. lntegrator 1 includes the rotating disc 44, the ball race 71, and the output shaft 72. Since the velocity at which any point on the disc 44 is rotating is directly proportional to the speed at which the disc is turning as well as to the radial distance of the point from the center of the disc, the ball, race 71 will impart a rotational velocity to output shaft 72 that is proportional to both the velocity of the disc and the position of the ball race. Since the velocity of the disc is determined by h, and the position of ball race by D +D the output of Integrator l is proportional to h(D +D Similarly, by passing this output to lntegrater 2, wherein the position of ball race 73 is proportional to D -D the output at shaft 75 is proportional to h(D +Dc)(D Dc)=II(D D Value is read on counter 76, which is set so that the annular volume may be read directly in cubic feet. Counter 76 has a zero reset knob 78 so that the counter starts operation-from the zero datum level of Figure 1.

From the foregoing description itwill be noted that a complete logging operation would involve the following: The caliper -13 is lowered to the proper depth within the bore hole. Control knob 55 is then rotated so that the desired setting D is read. Rotation of knob 55 is effective to rotate shaft 50, worm gear 51, gear 52, and cylinder member 32. Because of the oppositely threaded portions 45 and 46, the arms 33 and 34 are separated so that their respective distances from the center point They maintain their relative separation, even though they will .both move in unison as the caliper 13 measures changing bore hole diameter, D The zero reset knob 64 is pushed inwardly so that bevel gear 63 engages bevel gear 25. Rotation of knob 64 is then effective to turn shaft 27. This pushes shaft 29 axially because it cannot rotate within the square bushing 36. Since the overriding shaft 32 is squared between shoulder 47 and end face 48, the shaft 32 slides through gear 52. and end face 26 of shaft 29 will separate contact 66 from contact 67. This will extinguish the light 69 so that the operator knows that the starting position of the ball race positioning carriage has been reached.

As the caliper 13 is pulled up through the bore hole, Selsyn motor 40 is actuated. and rotates the disc 44 of Integrator l at a velocity corresponding to the increasing h information. At the same time, Selsyn motor 19 rotates the shaft 27, in one direction and then in the other depending on whether D is increasing or decreasing. This moves shaft 32 axially so that ball race positioning arms 37 and 38 move radially inwardly or outwardly with respect to the integrator discs 44 and 53. The arms move the ball races 71 and 73 by the same increment of radial distance, and in an amount proportional to D Since arm 37 was preset to an amount corresponding to +D the output of lntegrator l, at shaft 72 will be proportional to h(D +D This output is fed directly to disc 53 of Integrator 2, and since the arm 38 was preset according to -D the output of Integrator 2 will be proportional to This value is fed to the counter 76, which is arranged to read the differential annular volume directly in cubic feet.

It should be understood that although the computer has special application in the determination of annular volume, various changes can be made to give it even wider application. For example, if the pitch of the threaded portion 46 were double that of the threaded portion 45 of shaft 32, the computer would solve equations involving (x+2a)(xa). Similarly if the pitch of threaded portion 45 were double that of portion 46 q x-l-a) (x-2a) Or if the'arm 34 were contiguous with arm 33, the com 'puter would solve equations: involving (x-l-a) or (Ar-a). If other pitch ratios were used, other similar equations could be solved. Another possibility would be the use of the inventive principle with more than two integrators. Such an arrangement might be used to solve cubic, quartic, or other higher order equations involving the product of a number of factors such as (x-l-a), (x-a), (x-l-2a),etc.

Various other modificationsmay be suggested to one skilled in the art, but the true scope of the invention is to be determined by the appended claims.

Insofar as the claims are concerned, it should be understood that casing diameter designates the outer diameter of the casing. Likewise casing volume is the volume displaced by a cylinder having the same outer diameter as the casing.

What is claimed is:

'1. A computer comprising two integrators, each of said integrators having a rotating 'disc, a ball race engageable with and movable radially of the disc, and an output shaft engageable-with the ball race, means for rotating the disc'of the first integrator at a rate proportional to a first variable, a'positioner for'the ball. race of the first integrator effectiveto set the ball race at a position proportional to a second variable plus a constant, means linking the output shaft of the first integrator to the rotating disc of the second integrator, and means operable by said positioner to set the ball race of the second integrator at a position proportional to the said second variable minus the said constant, the output of the computer being'proportional to the product of the first variable times the difference between the squares of the second variable and the'said'constant.

2. A computer comprising two integrators, each of said integrators having a rotating disc, a ball race engageable with and movable radially of the disc, and an output shaft engageable with the ball race, means for rotating the disc of the first integrator at a rate proportional to a first variable, a positioner for the ball race of the first integrator effective to set the ball race at a position proportional to a second variable plus a constant, means linking the output shaft of the first integrator to the rotating disc of the second integrator, and means operable by said positioner to set the ball race of the second integrator at a position proportional to the said second variable minus the said constant, the output of the computer being proportional to the product of the first variable times the difference between the squares of the second variable and the said constant, said positioner including a cylinder having a left handed thread for a portion of its length and a right handed thread for a portion of its length, two arms carried on said cylinder and engageable respectively with the two threaded portions, whereby rotation ofthe cylinder is effective to move the arms toward or away from each other, said arms respectively being connected to said ball races for effecting setting of the latter.

3. A computer as in claim 2 wherein a control shaft is in driving engagement with the cylinder and causes rotation thereof, and including a metering knob on the control shaft so that the operating shaft can be given a desired set.

4. A computer arrangement adapted to be used in the logging of the annular volume between a length of casing and the walls of a bore hole comprising, means to caliper the varying bore hole diameter, means to measure the height of the hole with respect to a predetermined zero datum level, and means to correlate said height with the bore diameter at corresponding levels including means to continuously integrate the sum of the casing diameter and the bore hole diameter and to multiply it by the said height, and means operable by said integrating means and including means responsive to said height measuring means to subtract out the volume of casing, whereby a reading of the said annular volume is obtained.

5. A computer arrangement adapted to be used in the logging of the annular volume between a length of casing and the walls of a bore hole comprising, means to caliper the varying bore hole diameter, a first control member, means to preset the first control member according to a constant proportional to casing diameter, means to actuate the first control member so that its position is proportional to varying bore hole diameter increased by the said constant, a second control member, means to preset the second control member according to a constant proportional to the casing diameter, means to actuate the second control member so that its position is proportional to varying bore hole diameter decreased by the said constant, means to measure the height of the bore hole from a predetermined datum level, means to multiply the height measurements times bore hole diameter measurements as modified by the said constant, so that the product is proportional to the said annular volume.

6; Apparatus for determining the annular volume of a well borehole between the borehole wall and casing to be set in the borehole, comprising in combination, a well borehole calipering device lowerable into the borehole, said calipering device having arms engageable with the borehole wall, means for conducting an electrical signal from said calipering device proportional to the diameter of said borehole, means for raising said calipering device through the borehole, a first integrator disc, means for rotating said first integrator disc at a rate proportional to the height of travel of said calipering device, a second integrator disc, first variable control means including a source of power operated by said electrical signal from said calipering device for effecting rotation of said second integrator disc at a rate proportional to h(D +D wherein his the height of the calipering device in said borehole, D is the diameter of said borehole and D is the diameter of casing to be set in said borehole, an output member, and second variable control means for driving said output member at a value proportional to h(D +D )(D -D =h(D D and means driven by said output member for indicating the annular volume aforesaid in cubic feet.

7. Apparatus as defined in claim 6, wherein said first control means includes a member driven by said first integrator disc and means for adjusting the position of said driven member relative to said first integrator disc responsive to variations in the electrical signal from said calipering device, said second control means including a member driven by said second integrator disc and means for positioning said latter driven member relative to said second integrator, disc responsive to variations in the electrical signal from said calipering device.

References Cited in the file of this patent UNITED STATES PATENTS 2,194,477 Maxson et al Mar. 26, 1940 2,196,996 Lang Apr. 16, 1940 2,714,328 Hamburger et al. Aug. 2, 1955 2,716,340 Nance et al Aug. 30, 1955 OTHER REFERENCES Electronic Instruments by Greenwood et al., Radiation Laboratory Series #21, McGraw-Hill, 1948, pertinent pages 167-191.

I Product Engineering, vol. 20, November 1949, pages 122 and 123. 

